Systems of Physical Units. 87 



we obtain the relation 



unit length . 



unit velocity = 



unit time 



;-.',., , ,-, ,. . unit length ., ,. -, 



?. e. unit velocity and the quotient — ., , . ° — are identical 

 1 unit time 



magnitudes ; and we may therefore regard this quotient as the 



name of the unit of velocity, if this unit is derived from the 



units of length and time. But if we take the metre as the unit 



of length and the second as the unit of time, then the velocity 



derived from these units must be called T , which means 



second' 



that a velocity equal to 10 such units of velocity is to be 



.„ 1A metre 

 written 10 



second' 



The function — . . ,'. ° — is called bv Maxwell* the dimen- 

 unit time 



sions of the unit of velocity. If in all the fundamental equa- 

 tions we replace quantities by their units, we obtain a system 

 of equations according to which the absolute units are functions 

 amongst themselves of certain fundamental units. These 

 functions, which are obtained by solution of these last-named 

 equations, are the dimensions or names of the derived units. 



For the sake of brevity we will, as a rule, denote the units 

 by the same symbols as the quantities ; but, to avoid confu- 

 sion, we will include the symbols of units in brackets. Thus 

 [l~] will denote the unit of length. 



As an example of the different systems of units and dimen- 

 sions, we will consider the group of the six conceptions of 

 length, time, velocity, acceleration, mass, and force. We will 

 choose \J], [f\, and [m] as fundamental units ; then from the 

 four equations (2), (3), (6), (7) we may omit three constants, 

 say those of the first three equations ; and so we obtain as 

 fundamental equations the equations (8), (9), (12), (7). By 

 substitution of the units we obtain the following relations : — 



M=[j]> M=[*]. WK-4 [*]=[*£]. (15) 



By solution of these equations we obtain the following dimen- 

 sions: — 



M=[']> M-fil. W-g], W-KJ-OT 



If we keep the same fundamental equations, but choose [I]. 



* 'Treatise on Electricity and Magnetism/ i. p. 2, Compare Kohl- 

 rausch, Leitfaden, p. 207. 



